日本語学に興味が無い日本語ネイティブの場合は「熟字訓」という言葉を知らなくても「嗚呼(ああ)」とか「氷菓子(アイスクリーム)」という言葉を見たら納得して読みます。
I think that most Japanese can use 熟字訓 without knowing the name or method of reading.
But I guess many 熟字訓 have being forgotten among Japanese young people today.
For example young people usually call ice cream as just アイス.
Also, some people who wants to use unique nuance can use 嗚呼.
I have read there're about three thousands of 熟字訓, but many Japanese do know less than 100 I guess.
For example, 山葵 is written as わさび in many case today.